First, capacitors have a low impedance to AC signals, and keeps decreasing as as freq increases, and a high impedance do DC signals. When you put the shunt capacitor in place, you create a current divider network that basically absorbs the AC component of the voltage signal, thus leaving only the DC component.

EDIT: Note that current divides proportionally in such a way that most current flows through the path of least resistance. This explains why AC current is "absorbed" by the shunt capacitor.

The other is that since capacitors refuse to instantaneously change their voltage, they tend to smooth out fast voltage transitions, in this case, the ones occurring in the half/full rectified signals present after the rectifying diodes, once again creating a more DC like signal.